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# number theory

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The number 15 factorial ends in p zeros when converted to base 9. Find p

Oct 16, 2020

#1
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15 factorial is 1307674368000 (thx Google) under base 10. When converted to base 12, it turns into 191529600000.

There are 5 zeroes.

Oct 16, 2020
#2
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15! = 1307674368000 base 10 but you do not need to know this

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I can work out that there will be 3 zeros in base 10 quite easily.  All I need to know is how many factors of 10 are there.

10=2*5 so I am going to pull out all the 2 and 5 factors.

1*2*3*4*5*6*7*8*9*10*11*12*13*14*15

2   22 5 2    222  25       22       2    5 = 2^12 *  5^3  =  10^3 *2^9

so 15! can be divided by 10 three times so there will be 3 zeros in base 10

Which is what the calc gave me in the first place.

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To find how many zeros there will be base 9 then you have to work out how many factors of 9 go into it.

9=3*3

So see how may threes go into it and half that number will be your answer for trailing zeros base 9

Oct 16, 2020